# Algorithms and Experimental Study for the Traveling Salesman Problem of Second Order. Gerold Jäger

Save this PDF as:

Size: px
Start display at page:

Download "Algorithms and Experimental Study for the Traveling Salesman Problem of Second Order. Gerold Jäger"

## Transcription

1 Algorithms and Experimental Study for the Traveling Salesman Problem of Second Order Gerold Jäger joint work with Paul Molitor University Halle-Wittenberg, Germany August 22, 2008

2 Overview 1 Introduction Traveling Salesman Problem of Second Order Applications Related Problems Complexity Results 2 Heuristics for the Traveling Salesman Problem of Second Order Cheapest-Insert Algorithm Nearest-Neighbor Algorithm Two-Directional-Nearest-Neighbor Algorithm General-Assignment-Patching Algorithm k-opt Algorithm

3 Overview 3 Exact Algorithms for the Traveling Salesman Problem of Second Order Branch-and-Bound Algorithm Integer-Programming Algorithm 4 Experimental Comparison of Algorithms Experiments Experimental Comparison of Heuristics Experimental Comparison of Exact Algorithms

4 Introduction Traveling Salesman Problem of Second Order Traveling Salesman Problem of Second Order (2-TSP) Input: Output: Complete directed graph with n vertices v 1,..., v n. Three-dimensional cost matrix C = (c ijk ) R n,n,n with costs c ijk of a sequence of vertices (v i, v j, v k ). Complete tour T = (v i1,..., v in, v i1 ), Tour costs c(t ) = c in 1,i n,i 1 + c in,i 1,i 2 + n 2 l=1 c i l,i l+1,i l+2 are minimum.

5 Introduction Applications Bioinformatics: Models for transcription factor binding sites in gene regulation. Two models are the Permuted Markow (PM) Model and the Permuted Variable Length Markow (PVLM) Model. Solving order 1 of the PM/PVLM Model requires the solution of an ATSP instance. Solving order 2 of the PM/PVLM Model requires the solution of a 2-TSP instance.

6 Introduction Related Problems Asymmetric Traveling Salesman Problem (ATSP) Input: Output: Complete directed graph with n vertices v 1,..., v n. Cost matrix C = (c ij ) R n,n with costs c ij for an arc (v i, v j ). Complete tour T = (v i1,..., v in, v i1 ), Tour costs c(t ) = c in,i 1 + n 1 l=1 c i l,i l+1 are minimum.

7 Introduction Related Problems Assignment Problem (AP) Input: Output: Complete directed graph with n vertices v 1,..., v n. Cost matrix C = (c ij ) R n,n with costs c ij for an arc (v i, v j ). Set of cycles Z 1,..., Z k, where each vertex is visited exactly once. Sum of costs of all cycles c(t ) = c(z 1 ) + + c(z k ) is minimum, where cycle Z j = (v 1,..., v t ) has costs c(z j ) = c it,i 1 + t 1 l=1 c i l,i l+1.

8 Introduction Related Problems Assignment Problem of Second Order (2-AP) Input: Output: Complete directed graph with n vertices v 1,..., v n. Three-dimensional cost matrix C = (c ijk ) R n,n,n with costs c ijk of a sequence of vertices (v i, v j, v k ). Set of cycles Z 1,..., Z k, where each vertex is visited exactly once. Sum of costs of all cycles c(t ) = c(z 1 ) + + c(z k ) is minimum, where cycle Z j = (v 1,..., v t ) has costs c(z j ) = c it 1,i t,i 1 + c it,i 1,i 2 + t 2 l=1 c i l,i l+1,i l+2.

9 Introduction Complexity Results ATSP is NP-hard. [Karp, 1972] 2-TSP is NP-hard. Reduction from ATSP. AP is solvable in O(n 3 ). Hungarian Method [Kuhn, 1955] 2-AP is NP-hard. [Fischer, Lau, 2008]

10 Heuristics for the Traveling Salesman Problem of Second Order Eight heuristics are described in the paper. In the following five of them are presented.

11 Heuristics for the Traveling Salesman Problem of Second Order Cheapest-Insert Algorithm Generalization of an ATSP Algorithm of [Rosenkrantz, Stearns, Lewis, 1977]. Start with a good arc e = (v, w), and consider the subtour T := (v, w, v). Insert a good new vertex in T at a good place of the tour T. Repeat the last step, until T contains all vertices.

12 Heuristics for the Traveling Salesman Problem of Second Order Nearest-Neighbor Algorithm Generalization of an ATSP Algorithm of [Rosenkrantz, Stearns, Lewis, 1977]. Start with a good arc e = (v, w), and consider the path P = (v, w). Add a good new vertex to the tail of P, until P contains all vertices. Repeat the last step, until P contains all vertices. Close the path P, and receive a tour T.

13 Heuristics for the Traveling Salesman Problem of Second Order Two-Directional-Nearest-Neighbor Algorithm Generalization of the Nearest-Neighbor Algorithm. Vertices are added in both directions. At each step the direction is added in such a way that the difference between the second-best and the best vertex in each direction is larger, not from the best vertex over both directions.

14 Heuristics for the Traveling Salesman Problem of Second Order General-Assignment-Patching Algorithm Generalization of an ATSP Algorithm of [Karp, Steele, 1985]. Start with a good set of cycles Z 1,..., Z k, such that each vertex is visited exactly once. Which set is chosen? The set of cycles of a 2-AP solution is the set with the smallest costs. Problem: 2-AP is NP-hard. Idea: Compute a lower bound for the solution of the 2-AP instance by the solution of a corresponding AP instance. Solvable in O(n 3 ).

15 Heuristics for the Traveling Salesman Problem of Second Order General-Assignment-Patching Algorithm Choose two cycles Z 1 and Z 2 with the most vertices. Replace two arcs (v 1, w 1 ) Z 1 and (v 2, w 2 ) Z 2 by (v 1, w 2 ) and (v 2, w 1 ).

16 Heuristics for the Traveling Salesman Problem of Second Order General-Assignment-Patching Algorithm The arcs are chosen in such a way, that (v 1, w 1 ), (v 2, w 2 ) is as bad as possible and (v 1, w 2 ), (v 2, w 1 ) is as good as possible. Receive from Z 1 and Z 2 a new cycle Z 3. Repeat these steps, until you have only one cycle, which is a tour.

17 Heuristics for the Traveling Salesman Problem of Second Order k-opt Algorithm Generalization of an ATSP Algorithm of [Lin, Kernighan, 1973]. All previously presented algorithms are construction heuristics, i.e., they construct a tour. The k-opt Algorithm is an improvement heuristic, i.e., it starts with a tour (received by a construction heuristic) and improves it. A k-opt step changes a given tour T in such a way, that k tour arcs are replaced by k non-tour arcs and the set of arcs after the change is still a tour. The k-opt Algorithm applies tour improving k-opt steps with r k to a starting tour as long as such steps exist.

18 Exact Algorithms for the Traveling Salesman Problem of Second Order Branch-and-Bound Algorithm Compute with a heuristic a good upper bound for the instance. Go through all (n 1)! possible tours in lexicographical order. If an improvement occurs, the current optimum tour and the upper bound are updated. Go through all subpaths, first through subpaths with smaller number of vertices. Contract the subpath to a vertex. Compute the corresponding approximated 2-AP solution to the subpath. This solutions gives a local lower bound.

19 Exact Algorithms for the Traveling Salesman Problem of Second Order Branch-and-Bound Algorithm If the local lower bound of a subpath is not smaller than the current upper bound, all tours and subpaths, starting with this subpath, can be omitted from consideration.

20 Exact Algorithms for the Traveling Salesman Problem of Second Order Integer-Programming Algorithm 2-AP can be solved by the following IP model: n n n min c ijk x ijk (1) i=1,i j,k j=1,j k k=1 x ijk {0; 1} 1 i, j, k n, i j, i k, j k (2) n n x ijk = 1 1 k n (3) i=1,i j,k j=1,j k n n i=1,i j,k k=1,k j n n j=1,j i,k k=1,k j x ijk = 1 1 j n (4) x ijk = 1 1 i n (5)

21 Exact Algorithms for the Traveling Salesman Problem of Second Order Integer-Programming Algorithm n k=1,k i,k j x ijk = k=1,k i,k j x kij 1 i j n (6) For solving not only the 2-AP, but also the 2-TSP, all possible subtours have to be forbidden. For the subtour (v s1,..., v st, v s1 ) the following inequality has to be added: t 2 x st 1,s t,s 1 + x st,s 1,s 2 + x si,s i+1,s i+2 t 1 (7) i=1 Unfortunately an exponential number of such subtours exists.

22 Exact Algorithms for the Traveling Salesman Problem of Second Order Integer-Programming Algorithm Therefore at each step only the necessary subtour inequalities are added: 1 Define IP by the conditions (1), (2), (3), (4), (5), (6). 2 Solve IP. 3 Receive a solution with k cycles Z 1,..., Z k and cycle Z = (v s1,..., v st, v s1 ) with the minimum number of vertices t. 4 IF k = 1 5 THEN TERMINATE with solution Z. 6 ELSE Add condition (7) for cycle Z to IP. 7 GOTO 2. If an IP solver (as CPLEX) is started with a good upper bound (received by a heuristic), the solver can be accelerated.

23 Experimental Comparison of Algorithms Experiments We compare all basic heuristics and all basic heuristics followed by a 5-OPT step in quality (upper bound) and time. We compare the upper bound of all basic and OPT heuristics with the optima. We compare all exact algorithms in time. All experiments were done for two classes: real and random instances.

24 Experimental Comparison of Algorithms Experimental Comparison of Heuristics The times of the OPT versions are much larger than for the basic versions. The times of all basic versions are similar. The times of all OPT versions are similar. The quality of the OPT versions is much better than for the basic versions. The quality results for the random and real instances completely differ: For the real instances, the OPT Cheapest-Insert Algorithm and the OPT General Assignment Patching Algorithm are the best algorithms. For the random instances, the OPT Two-Directional- Nearest-Neighbor Algorithm is the best algorithm.

25 Experimental Comparison of Algorithms Experimental Comparison of Heuristics Real instances: for 44 of 45 instances at least one (of seven) OPT heuristics finds the optimum. Random instances: the upper bounds are more far away from the optimum.

26 Experimental Comparison of Algorithms Experimental Comparison of Exact Algorithms Random instances can be solved faster than real instances. Random instances and real instances of smaller dimensions: BnB Algorithm faster. Real instances of larger dimensions (21 and 26 ): Only IP-Algorithm successful.

27 Experimental Comparison of Algorithms Experimental Comparison of Exact Algorithms Thanks for your attention!

### Combinatorial Optimization - Lecture 14 - TSP EPFL

Combinatorial Optimization - Lecture 14 - TSP EPFL 2012 Plan Simple heuristics Alternative approaches Best heuristics: local search Lower bounds from LP Moats Simple Heuristics Nearest Neighbor (NN) Greedy

### Traveling Salesman Problem (TSP) Input: undirected graph G=(V,E), c: E R + Goal: find a tour (Hamiltonian cycle) of minimum cost

Traveling Salesman Problem (TSP) Input: undirected graph G=(V,E), c: E R + Goal: find a tour (Hamiltonian cycle) of minimum cost Traveling Salesman Problem (TSP) Input: undirected graph G=(V,E), c: E R

### Assignment 3b: The traveling salesman problem

Chalmers University of Technology MVE165 University of Gothenburg MMG631 Mathematical Sciences Linear and integer optimization Optimization with applications Emil Gustavsson Assignment information Ann-Brith

### Traveling Salesman Problem. Algorithms and Networks 2014/2015 Hans L. Bodlaender Johan M. M. van Rooij

Traveling Salesman Problem Algorithms and Networks 2014/2015 Hans L. Bodlaender Johan M. M. van Rooij 1 Contents TSP and its applications Heuristics and approximation algorithms Construction heuristics,

### 1 The Traveling Salesperson Problem (TSP)

CS 598CSC: Approximation Algorithms Lecture date: January 23, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im In the previous lecture, we had a quick overview of several basic aspects of approximation

### Last topic: Summary; Heuristics and Approximation Algorithms Topics we studied so far:

Last topic: Summary; Heuristics and Approximation Algorithms Topics we studied so far: I Strength of formulations; improving formulations by adding valid inequalities I Relaxations and dual problems; obtaining

### Dynamic programming. Trivial problems are solved first More complex solutions are composed from the simpler solutions already computed

Dynamic programming Solves a complex problem by breaking it down into subproblems Each subproblem is broken down recursively until a trivial problem is reached Computation itself is not recursive: problems

### Introduction to Approximation Algorithms

Introduction to Approximation Algorithms Dr. Gautam K. Das Departmet of Mathematics Indian Institute of Technology Guwahati, India gkd@iitg.ernet.in February 19, 2016 Outline of the lecture Background

### Improving the Efficiency of Helsgaun s Lin-Kernighan Heuristic for the Symmetric TSP

Improving the Efficiency of Helsgaun s Lin-Kernighan Heuristic for the Symmetric TSP Dirk Richter 1, Boris Goldengorin 2,3,4, Gerold Jäger 5, and Paul Molitor 1 1 Computer Science Institute, University

### Multi-Objective Combinatorial Optimization: The Traveling Salesman Problem and Variants

Multi-Objective Combinatorial Optimization: The Traveling Salesman Problem and Variants C. Glaßer 1 C. Reitwießner 1 H. Schmitz 2 M. Witek 1 1 University of Würzburg, Germany 2 Trier University of Applied

### Modified Order Crossover (OX) Operator

Modified Order Crossover (OX) Operator Ms. Monica Sehrawat 1 N.C. College of Engineering, Israna Panipat, Haryana, INDIA. Mr. Sukhvir Singh 2 N.C. College of Engineering, Israna Panipat, Haryana, INDIA.

### Module 6 P, NP, NP-Complete Problems and Approximation Algorithms

Module 6 P, NP, NP-Complete Problems and Approximation Algorithms Dr. Natarajan Meghanathan Associate Professor of Computer Science Jackson State University Jackson, MS 39217 E-mail: natarajan.meghanathan@jsums.edu

### COMP 355 Advanced Algorithms Approximation Algorithms: VC and TSP Chapter 11 (KT) Section (CLRS)

COMP 355 Advanced Algorithms Approximation Algorithms: VC and TSP Chapter 11 (KT) Section 35.1-35.2(CLRS) 1 Coping with NP-Completeness Brute-force search: This is usually only a viable option for small

### Kurt Mehlhorn, MPI für Informatik. Curve and Surface Reconstruction p.1/25

Curve and Surface Reconstruction Kurt Mehlhorn MPI für Informatik Curve and Surface Reconstruction p.1/25 Curve Reconstruction: An Example probably, you see more than a set of points Curve and Surface

### A Tabu Search Heuristic for the Generalized Traveling Salesman Problem

A Tabu Search Heuristic for the Generalized Traveling Salesman Problem Jacques Renaud 1,2 Frédéric Semet 3,4 1. Université Laval 2. Centre de Recherche sur les Technologies de l Organisation Réseau 3.

### Improving the Held and Karp Approach with Constraint Programming

Improving the Held and Karp Approach with Constraint Programming Pascal Benchimol 1, Jean-Charles Régin 2, Louis-Martin Rousseau 1, Michel Rueher 2, Willem-Jan van Hoeve 3 1 CIRRELT,École Polytechnique

### Lagrangian Relaxation in CP

Lagrangian Relaxation in CP Willem-Jan van Hoeve CPAIOR 016 Master Class Overview 1. Motivation for using Lagrangian Relaxations in CP. Lagrangian-based domain filtering Example: Traveling Salesman Problem.

### JOURNAL OF OBJECT TECHNOLOGY

JOURNAL OF OBJECT TECHNOLOGY Online at http://www.jot.fm. Published by ETH Zurich, Chair of Software Engineering JOT, 2005 Vol. 4, No. 1, January-February 2005 A Java Implementation of the Branch and Bound

### A NEW HEURISTIC ALGORITHM FOR MULTIPLE TRAVELING SALESMAN PROBLEM

TWMS J. App. Eng. Math. V.7, N.1, 2017, pp. 101-109 A NEW HEURISTIC ALGORITHM FOR MULTIPLE TRAVELING SALESMAN PROBLEM F. NURIYEVA 1, G. KIZILATES 2, Abstract. The Multiple Traveling Salesman Problem (mtsp)

### Algorithms for the Precedence Constrained Generalized Travelling Salesperson Problem

MASTER S THESIS Algorithms for the Precedence Constrained Generalized Travelling Salesperson Problem RAAD SALMAN Department of Mathematical Sciences Division of Mathematics CHALMERS UNIVERSITY OF TECHNOLOGY

### Algorithm Design and Analysis

Algorithm Design and Analysis LECTURE 29 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/7/2016 Approximation

### 3 INTEGER LINEAR PROGRAMMING

3 INTEGER LINEAR PROGRAMMING PROBLEM DEFINITION Integer linear programming problem (ILP) of the decision variables x 1,..,x n : (ILP) subject to minimize c x j j n j= 1 a ij x j x j 0 x j integer n j=

### to the Traveling Salesman Problem 1 Susanne Timsj Applied Optimization and Modeling Group (TOM) Department of Mathematics and Physics

An Application of Lagrangian Relaxation to the Traveling Salesman Problem 1 Susanne Timsj Applied Optimization and Modeling Group (TOM) Department of Mathematics and Physics M lardalen University SE-721

### A SAT Based Effective Algorithm for the Directed Hamiltonian Cycle Problem

A SAT Based Effective Algorithm for the Directed Hamiltonian Cycle Problem Gerold Jäger and Weixiong Zhang 2 Computer Science Institute Christian-Albrechts-University of Kiel D-248 Kiel, Germany E-mail:

### Combining Two Local Searches with Crossover: An Efficient Hybrid Algorithm for the Traveling Salesman Problem

Combining Two Local Searches with Crossover: An Efficient Hybrid Algorithm for the Traveling Salesman Problem Weichen Liu, Thomas Weise, Yuezhong Wu and Qi Qi University of Science and Technology of Chine

### Networks: Lecture 2. Outline

Networks: Lecture Amedeo R. Odoni November 0, 00 Outline Generic heuristics for the TSP Euclidean TSP: tour construction, tour improvement, hybrids Worst-case performance Probabilistic analysis and asymptotic

### HEURISTIC ALGORITHMS FOR THE SINGLE VEHICLE DIAL-A-RIDE PROBLEM

Journal of the Operations Research Society of Japan Vol. 33, No. 4, December 1990 1990 The Operations Research Society of Japan HEURISTIC ALGORITHMS FOR THE SINGLE VEHICLE DIAL-A-RIDE PROBLEM Mikio Kubo

### Vertex Cover Approximations

CS124 Lecture 20 Heuristics can be useful in practice, but sometimes we would like to have guarantees. Approximation algorithms give guarantees. It is worth keeping in mind that sometimes approximation

### Chapter 9 Graph Algorithms

Introduction graph theory useful in practice represent many real-life problems can be if not careful with data structures Chapter 9 Graph s 2 Definitions Definitions an undirected graph is a finite set

### Hardware-Software Codesign

Hardware-Software Codesign 4. System Partitioning Lothar Thiele 4-1 System Design specification system synthesis estimation SW-compilation intellectual prop. code instruction set HW-synthesis intellectual

### GAUSSIAN VARIABLE NEIGHBORHOOD SEARCH FOR THE FILE TRANSFER SCHEDULING PROBLEM

Yugoslav Journal of Operations Research 26 (2016), Number 2, 173 188 DOI: 10.2298/YJOR150124006D GAUSSIAN VARIABLE NEIGHBORHOOD SEARCH FOR THE FILE TRANSFER SCHEDULING PROBLEM Zorica DRAŽIĆ Faculty of

### Parameterized Complexity - an Overview

Parameterized Complexity - an Overview 1 / 30 Parameterized Complexity - an Overview Ue Flarup 1 flarup@imada.sdu.dk 1 Department of Mathematics and Computer Science University of Southern Denmark, Odense,

### Improvement heuristics for the Sparse Travelling Salesman Problem

Improvement heuristics for the Sparse Travelling Salesman Problem FREDRICK MTENZI Computer Science Department Dublin Institute of Technology School of Computing, DIT Kevin Street, Dublin 8 IRELAND http://www.comp.dit.ie/fmtenzi

### Approximate Muscle Guided Beam Search for Three-Index Assignment Problem

Approximate Muscle Guided Beam Search for Three-Index Assignment Problem He Jiang, Shuwei Zhang, Zhilei Ren, Xiaochen Lai, and Yong Piao Software School, Dalian University of Technology, Dalian, 116621,

### Chapter 9 Graph Algorithms

Chapter 9 Graph Algorithms 2 Introduction graph theory useful in practice represent many real-life problems can be if not careful with data structures 3 Definitions an undirected graph G = (V, E) is a

### Comparison of TSP Algorithms

Comparison of TSP Algorithms Project for Models in Facilities Planning and Materials Handling December 1998 Participants: Byung-In Kim Jae-Ik Shim Min Zhang Executive Summary Our purpose in this term project

### Pre-requisite Material for Course Heuristics and Approximation Algorithms

Pre-requisite Material for Course Heuristics and Approximation Algorithms This document contains an overview of the basic concepts that are needed in preparation to participate in the course. In addition,

### 1 The Traveling Salesman Problem

Comp 260: Advanced Algorithms Tufts University, Spring 2011 Prof. Lenore Cowen Scribe: Jisoo Park Lecture 3: The Traveling Salesman Problem 1 The Traveling Salesman Problem The Traveling Salesman Problem

### OPERATIONS RESEARCH. Transportation and Assignment Problems

OPERATIONS RESEARCH Chapter 2 Transportation and Assignment Problems Prof Bibhas C Giri Professor of Mathematics Jadavpur University West Bengal, India E-mail : bcgirijumath@gmailcom MODULE-3: Assignment

### 3 No-Wait Job Shops with Variable Processing Times

3 No-Wait Job Shops with Variable Processing Times In this chapter we assume that, on top of the classical no-wait job shop setting, we are given a set of processing times for each operation. We may select

### Module 6 NP-Complete Problems and Heuristics

Module 6 NP-Complete Problems and Heuristics Dr. Natarajan Meghanathan Professor of Computer Science Jackson State University Jackson, MS 97 E-mail: natarajan.meghanathan@jsums.edu Optimization vs. Decision

### Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem

Journal of Artificial Intelligence Research 2 (24) 47-497 Submitted /3; published 4/4 Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem Weixiong Zhang Department of Computer

### Critical Node Detection Problem. Panos Pardalos Distinguished Professor CAO, Dept. of Industrial and Systems Engineering, University of Florida

Critical Node Detection Problem ITALY May, 2008 Panos Pardalos Distinguished Professor CAO, Dept. of Industrial and Systems Engineering, University of Florida Outline of Talk Introduction Problem Definition

### Heuristic Approaches to Solve Traveling Salesman Problem

TELKOMNIKA Indonesian Journal of Electrical Engineering Vol. 15, No. 2, August 2015, pp. 390 ~ 396 DOI: 10.11591/telkomnika.v15i2.8301 390 Heuristic Approaches to Solve Traveling Salesman Problem Malik

### Outline. Optimales Recycling - Tourenplanung in der Altglasentsorgung

1 Optimales Recycling - Ruhr-Universität Bochum, 15.02.2013 2 1. Introduction and Motivation 2. Problem Definition 3. Literature Review 4. Mathematical Model 5. Variable Neighborhood Search 6. Numerical

### Constructive and destructive algorithms

Constructive and destructive algorithms Heuristic algorithms Giovanni Righini University of Milan Department of Computer Science (Crema) Constructive algorithms In combinatorial optimization problems every

### An iteration of Branch and Bound One iteration of Branch and Bound consists of the following four steps: Some definitions. Branch and Bound.

ranch and ound xamples and xtensions jesla@man.dtu.dk epartment of Management ngineering Technical University of enmark ounding ow do we get ourselves a bounding function? Relaxation. Leave out some constraints.

### Computing Crossing-Free Configurations with Minimum Bottleneck

Computing Crossing-Free Configurations with Minimum Bottleneck Sándor P. Fekete 1 and Phillip Keldenich 1 1 Department of Computer Science, TU Braunschweig, Germany {s.fekete,p.keldenich}@tu-bs.de Abstract

### Improved Filtering for Weighted Circuit Constraints

Constraints manuscript No. (will be inserted by the editor) Improved Filtering for Weighted Circuit Constraints Pascal Benchimol Willem-Jan van Hoeve Jean-Charles Régin Louis-Martin Rousseau Michel Rueher

### Construction heuristics for the asymmetric TSP

European Journal of Operational Research 129 (2001) 555±568 www.elsevier.com/locate/dsw Theory and Methodology Construction heuristics for the asymmetric TSP Fred Glover a, Gregory Gutin b, *, Anders Yeo

### Rollout Algorithms for Discrete Optimization: A Survey

Rollout Algorithms for Discrete Optimization: A Survey by Dimitri P. Bertsekas Massachusetts Institute of Technology Cambridge, MA 02139 dimitrib@mit.edu August 2010 Abstract This chapter discusses rollout

### The Vehicle Routing Problem with Time Windows

The Vehicle Routing Problem with Time Windows Dr Philip Kilby Team Leader, Optimisation Applications and Platforms June 2017 www.data61.csiro.au Outline Problem Description Solving the VRP Construction

### A Polynomial-Time Deterministic Approach to the Traveling Salesperson Problem

A Polynomial-Time Deterministic Approach to the Traveling Salesperson Problem Ali Jazayeri and Hiroki Sayama Center for Collective Dynamics of Complex Systems Department of Systems Science and Industrial

### Notes for Recitation 9

6.042/18.062J Mathematics for Computer Science October 8, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 9 1 Traveling Salesperson Problem Now we re going to talk about a famous optimization

### Questions? You are given the complete graph of Facebook. What questions would you ask? (What questions could we hope to answer?)

P vs. NP What now? Attribution These slides were prepared for the New Jersey Governor s School course The Math Behind the Machine taught in the summer of 2011 by Grant Schoenebeck Large parts of these

### Algorithm Design Techniques (III)

Algorithm Design Techniques (III) Minimax. Alpha-Beta Pruning. Search Tree Strategies (backtracking revisited, branch and bound). Local Search. DSA - lecture 10 - T.U.Cluj-Napoca - M. Joldos 1 Tic-Tac-Toe

### The Traveling Salesperson Problem with Forbidden Neighborhoods on Regular 3D Grids

The Traveling Salesperson Problem with Forbidden Neighborhoods on Regular 3D Grids Anja Fischer, Philipp Hungerländer 2, and Anna Jellen 2 Tehnische Universität Dortmund, Germany, anja2.fischer@tu-dortmund.de,

### Solution of P versus NP problem

Algorithms Research 2015, 4(1): 1-7 DOI: 105923/jalgorithms2015040101 Solution of P versus NP problem Mustapha Hamidi Meknes, Morocco Abstract This paper, taking Travelling Salesman Problem as our object,

### An Adaptive k-opt Method for Solving Traveling Salesman Problem

An Adaptive k-opt Method for Solving Traveling Salesman Problem Zhibei Ma, Lantao Liu, Gaurav S. Sukhatme Abstract This paper presents a new heuristic solution to the traveling salesman problem (TSP).

### General k-opt submoves for the Lin Kernighan TSP heuristic

Math. Prog. Comp. (2009) 1:119 163 DOI 10.1007/s12532-009-0004-6 FULL LENGTH PAPER General k-opt submoves for the Lin Kernighan TSP heuristic Keld Helsgaun Received: 25 March 2009 / Accepted: 3 June 2009

### Genetic algorithms for the traveling salesman problem

337 Genetic Algorithms Annals of Operations Research 63(1996)339-370 339 Genetic algorithms for the traveling salesman problem Jean-Yves Potvin Centre de Recherche sur les Transports, Universitd de Montrgal,

### 1 Better Approximation of the Traveling Salesman

Stanford University CS261: Optimization Handout 4 Luca Trevisan January 13, 2011 Lecture 4 In which we describe a 1.5-approximate algorithm for the Metric TSP, we introduce the Set Cover problem, observe

### Effective Local and Guided Variable Neighbourhood Search Methods for the Asymmetric Travelling Salesman Problem

Effective Local and Guided Variable Neighbourhood Search Methods for the Asymmetric Travelling Salesman Problem Edmund K. Burke, Peter I. Cowling, and Ralf Keuthen Automated Scheduling, Optimization, and

### Time dependent optimization problems in networks

Gonny Hauwert Time dependent optimization problems in networks Master thesis, defended on November 10, 2010 Thesis advisors: Dr. F.M. Spieksma and Prof. Dr. K.I. Aardal Specialisation: Applied Mathematics

### Graph Algorithms Matching

Chapter 5 Graph Algorithms Matching Algorithm Theory WS 2012/13 Fabian Kuhn Circulation: Demands and Lower Bounds Given: Directed network, with Edge capacities 0and lower bounds l for Node demands for

### LKH User Guide Version 2.0 (November 2007)

LKH User Guide Version 2.0 (November 2007) by Keld Helsgaun E-mail: keld@ruc.dk 1. Introduction The Lin-Kernighan heuristic [1] is generally considered to be one of the most successful methods for generating

### LKH User Guide. Version 1.3 (July 2002) by Keld Helsgaun

LKH User Guide Version 1.3 (July 2002) by Keld Helsgaun E-mail: keld@ruc.dk 1. Introduction The Lin-Kernighan heuristic [1] is generally considered to be one of the most successful methods for generating

### Fast Hierarchical Clustering via Dynamic Closest Pairs

Fast Hierarchical Clustering via Dynamic Closest Pairs David Eppstein Dept. Information and Computer Science Univ. of California, Irvine http://www.ics.uci.edu/ eppstein/ 1 My Interest In Clustering What

### Partitioning. Course contents: Readings. Kernighang-Lin partitioning heuristic Fiduccia-Mattheyses heuristic. Chapter 7.5.

Course contents: Partitioning Kernighang-Lin partitioning heuristic Fiduccia-Mattheyses heuristic Readings Chapter 7.5 Partitioning 1 Basic Definitions Cell: a logic block used to build larger circuits.

### The Bounded Edge Coloring Problem and Offline Crossbar Scheduling

The Bounded Edge Coloring Problem and Offline Crossbar Scheduling Jonathan Turner WUCSE-05-07 Abstract This paper introduces a variant of the classical edge coloring problem in graphs that can be applied

### 11/17/2009 Comp 590/Comp Fall

Lecture 20: Clustering and Evolution Study Chapter 10.4 10.8 Problem Set #5 will be available tonight 11/17/2009 Comp 590/Comp 790-90 Fall 2009 1 Clique Graphs A clique is a graph with every vertex connected

### 15-451/651: Design & Analysis of Algorithms November 4, 2015 Lecture #18 last changed: November 22, 2015

15-451/651: Design & Analysis of Algorithms November 4, 2015 Lecture #18 last changed: November 22, 2015 While we have good algorithms for many optimization problems, the previous lecture showed that many

### The Traveling Salesperson's Toolkit

The Traveling Salesperson's Toolkit Mark Michael, Ph.D. King's College, 133 N. River St. Wilkes-Barre, PA 18711, USA mmichael@gw02.kings.edu Introduction In theoretical computer science, NP-complete problems

### Journal of Graph Algorithms and Applications

Journal of Graph Algorithms and Applications http://www.cs.brown.edu/publications/jgaa/ vol. 1, no. 1, pp. 1 25 (1997) 2-Layer Straightline Crossing Minimization: Performance of Exact and Heuristic Algorithms

### 1. trees does the network shown in figure (a) have? (b) How many different spanning. trees does the network shown in figure (b) have?

2/28/18, 8:24 M 1. (a) ow many different spanning trees does the network shown in figure (a) have? (b) ow many different spanning trees does the network shown in figure (b) have? L K M P N O L K M P N

### Lecture Notes for IEOR 266: Graph Algorithms and Network Flows

Lecture Notes for IEOR 266: Graph Algorithms and Network Flows Professor Dorit S. Hochbaum Contents 1 Introduction 1 1.1 Assignment problem.................................... 1 1.2 Basic graph definitions...................................

### 3 Euler Tours, Hamilton Cycles, and Their Applications

3 Euler Tours, Hamilton Cycles, and Their Applications 3.1 Euler Tours and Applications 3.1.1 Euler tours Carefully review the definition of (closed) walks, trails, and paths from Section 1... Definition

### All 0-1 Polytopes are. Abstract. We study the facial structure of two important permutation polytopes

All 0-1 Polytopes are Traveling Salesman Polytopes L.J. Billera and A. Sarangarajan y Abstract We study the facial structure of two important permutation polytopes in R n2, the Birkho or assignment polytope

### Improving Lin-Kernighan-Helsgaun with Crossover on Clustered Instances of the TSP

Improving Lin-Kernighan-Helsgaun with Crossover on Clustered Instances of the TSP Doug Hains, Darrell Whitley, and Adele Howe Colorado State University, Fort Collins CO, USA Abstract. Multi-trial Lin-Kernighan-Helsgaun

### Innovative Systems Design and Engineering ISSN (Paper) ISSN (Online) Vol.5, No.1, 2014

Abstract Tool Path Optimization of Drilling Sequence in CNC Machine Using Genetic Algorithm Prof. Dr. Nabeel Kadim Abid Al-Sahib 1, Hasan Fahad Abdulrazzaq 2* 1. Thi-Qar University, Al-Jadriya, Baghdad,

### Dynamic Programming II

Lecture 11 Dynamic Programming II 11.1 Overview In this lecture we continue our discussion of dynamic programming, focusing on using it for a variety of path-finding problems in graphs. Topics in this

### P and NP (Millenium problem)

CMPS 2200 Fall 2017 P and NP (Millenium problem) Carola Wenk Slides courtesy of Piotr Indyk with additions by Carola Wenk CMPS 2200 Introduction to Algorithms 1 We have seen so far Algorithms for various

### Planar graphs: multiple-source shortest paths, brick decomposition, and Steiner tree

Planar graphs: multiple-source shortest paths, brick decomposition, and Steiner tree Philip Klein joint work with Glencora Borradaile and Claire Mathieu Program: For fundamental optimization problems on

### Approximating the Maximum Quadratic Assignment Problem 1

Approximating the Maximum Quadratic Assignment Problem 1 Esther M. Arkin Refael Hassin 3 Maxim Sviridenko 4 Keywords: Approximation algorithm; quadratic assignment problem 1 Introduction In the maximum

### An Ant Colony Optimization Algorithm for Solving Travelling Salesman Problem

1 An Ant Colony Optimization Algorithm for Solving Travelling Salesman Problem Krishna H. Hingrajiya, Ravindra Kumar Gupta, Gajendra Singh Chandel University of Rajiv Gandhi Proudyogiki Vishwavidyalaya,

### A HIGH PERFORMANCE ALGORITHM FOR SOLVING LARGE SCALE TRAVELLING SALESMAN PROBLEM USING DISTRIBUTED MEMORY ARCHITECTURES

A HIGH PERFORMANCE ALGORITHM FOR SOLVING LARGE SCALE TRAVELLING SALESMAN PROBLEM USING DISTRIBUTED MEMORY ARCHITECTURES Khushboo Aggarwal1,Sunil Kumar Singh2, Sakar Khattar3 1,3 UG Research Scholar, Bharati

### A Distributed Chained Lin-Kernighan Algorithm for TSP Problems

A Distributed Chained Lin-Kernighan Algorithm for TSP Problems Thomas Fischer Department of Computer Science University of Kaiserslautern fischer@informatik.uni-kl.de Peter Merz Department of Computer

### (a) (b) (c) Phase1. Phase2. Assignm ent offfs to scan-paths. Phase3. Determination of. connection-order offfs. Phase4. Im provem entby exchanging FFs

Scan-chain Optimization lgorithms for Multiple Scan-paths Susumu Kobayashi Masato Edahiro Mikio Kubo C&C Media Research Laboratories NEC Corporation Kawasaki, Japan Logistics and Information Engineering

### Vehicle Routing Heuristic Methods

DM87 SCHEDULING, TIMETABLING AND ROUTING Outline 1. Construction Heuristics for VRPTW Lecture 19 Vehicle Routing Heuristic Methods 2. Local Search 3. Metaheuristics Marco Chiarandini 4. Other Variants

### Computers & Operations Research

Computers & Operations Research 36 (2009) 2619 -- 2631 Contents lists available at ScienceDirect Computers & Operations Research journal homepage: www.elsevier.com/locate/cor Design and analysis of stochastic

### Traveling Salesperson Problem (TSP)

TSP-0 Traveling Salesperson Problem (TSP) Input: Undirected edge weighted complete graph G = (V, E, W ), where W : e R +. Tour: Find a path that starts at vertex 1, visits every vertex exactly once, and

### Discrete Optimization. Lecture Notes 2

Discrete Optimization. Lecture Notes 2 Disjunctive Constraints Defining variables and formulating linear constraints can be straightforward or more sophisticated, depending on the problem structure. The

### NP-Hardness. We start by defining types of problem, and then move on to defining the polynomial-time reductions.

CS 787: Advanced Algorithms NP-Hardness Instructor: Dieter van Melkebeek We review the concept of polynomial-time reductions, define various classes of problems including NP-complete, and show that 3-SAT

### Lecture 20: Satisfiability Steven Skiena. Department of Computer Science State University of New York Stony Brook, NY

Lecture 20: Satisfiability Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Problem of the Day Suppose we are given

### UML modeling for traveling salesman problem based on genetic algorithms

SOUTHEAST EUROPE JOURNAL OF SOFT COMPUTING Available online at www.scjournal.com.ba UML modeling for traveling salesman problem based on genetic algorithms Muzafer Saračević a, Sead Mašović b, Šemsudin

### Constructing arbitrarily large graphs with a specified number of Hamiltonian cycles

Electronic Journal of Graph Theory and Applications 4 (1) (2016), 18 25 Constructing arbitrarily large graphs with a specified number of Hamiltonian cycles Michael School of Computer Science, Engineering

### 9 Heuristic Methods for Combinatorial Optimization Problems

Contents 9 Heuristic Methods for Combinatorial Optimization Problems 425 9.1 WhatAreHeuristicMethods?... 425 9.2 WhyUseHeuristics?... 426 9.3 General Principles in DesigningHeuristicMethods... 431 9.4

### Polynomial-Time Approximation Algorithms

6.854 Advanced Algorithms Lecture 20: 10/27/2006 Lecturer: David Karger Scribes: Matt Doherty, John Nham, Sergiy Sidenko, David Schultz Polynomial-Time Approximation Algorithms NP-hard problems are a vast